1
Guru Nanak Dev Engineering College
Ludhiana
Department Of Civil Engineering
Topic Design philosophies
Submitted By Aryan Gupta
C.R.N - 2214006
Submitted To Dr H.S Rai Sir
DESIGN PHILOSOPHIES
Objectives of RCC Design
To achieve a structure that will result in a Safe and Economical solution. It
consists of the following steps:
1Idealization of Structure for Analysis [The process of replacing an actual
structure with a simple system conducive to analysis is called structural
idealization].
2Estimation of Loads
3 Analysis of Idealized Structure to determine
(a) Axial Thrust
(b) Shear
(c) Bending Moments
(d) Deflections
4 Design of Structural Elements
5 Material Specifications and Detailing of Reinforcement
6 Detailed Structural Drawing
Design Philosophies
1 Working Stress Method
2 Ultimate Load method
3 Limit State Design
BEAM
Idealized Beam
D
B
WORKING STRESS METHOD
Fig 1: Stress Strain Curves for Concrete and Steel
Working Stress Method is an Elastic Method i.e. the material behaves Elastically
and Hook's Law is valid i.e. Stress is Proportional to Strain.
This means Strength of only that part of the material is considered which is under
Elastic Limit i.e. up to point 'A'.
The remaining Strength Beyond point A is not taken into consideration for a
particular Load.
This can be summarized as
1 Concrete is Elastic
2 Steel and Concrete act together Elastically
3 The relationship between Stress and Strain is Linear right upto failure/collapse
of the structure.
4 The Permissible Stresses for Concrete and Steel are not exceeded anywhere in
the Structure when subjected to the worst combinations of Working Loads.
5 The Sections are Designed as per the Elastic Theory of Bending assuming that
both materials i.e. concrete and steel obey the Hook's law.
6 The Elastic Theory assumes a Linear Variation of Strain and Stress i.e. Zero at
the Neutral Axis and Maximum at the extreme fibre.
Fig 2 Stress & Strain Variation Along the Depth of
Beam
Beam Section
Strain Variation Stress Variation
Beam Section Strain Variation Stress Variation
ASSUMPTIONS
1A Section which is Plane before Bending remains Plain after Bending
2 Bond between Steel and Concrete is perfect within the Elastic Limit of Steel
3 The Stress Strain Relationship of Steel and Concrete under Working Loads is
a Straight Line
4 Concrete is Elastic i.e. Stress varies Linearly from Zero at the Neutral Axis to
a maximum at the extreme fibre.
5 The Modular Ratio 'm' has the value
Where:-
σcbc :-Permissible Compressive Strength due to Bending in MPa
Permissible Stresses
Although the Stresses are kept within the Elastic Limits and most of the
Strength is not taken into account, the Stresses in Concrete and Steel are further
reduced by a Factor of Safety to account for uncertainties in the Estimation of
the Working Loads and variations in the Properties of Materials
Factor of Safety
For Concrete 3
Steel 1.78
The Permissible Stress is obtained by dividing Grade of concrete by 3 and Yield
Strength of Steel be 1.78.
Different values of Permissible Stresses for Concrete and Steel are given in
Tables 21 and 22 IS 456 - 2000 and for Shear Stresses in Table 23 of IS 456 -
2000.
Hence for properly designed Structural Elements the Stresses computed under
the action of working loads will be well within the Elastic Range. e.g.
For M20 Concrete :
Permissible Compressive Stress Permissible Stress in Concrete
in Bending Compression
For Fe250 Steel :
Permissible Tensile Stress Permissible Stress in Concrete
in Bending Tension
Structural Element is Designed for :
(1) Strength and (2) Stiffness under working loads
Strength Consists of:
(a) Compression Bending
(b) Shear
(c) Torsion
(d) Combination of above factors
Stiffness Consists of:
(a) Deflections
(b) Cracking
(d) Vibrations
Types of Loads
(a) Dead Loads
(b) Live Loads or Imposed Loads
(c) Wind Load
(d) Earthquake Load
Static Loads
Dynamic Loads
DRAWBACKS
1 Concrete is not an Elastic Material. The inelastic material behaviour of
concrete actually starts from very low stresses. The Actual stress distribution
in concrete section is in fact Non Linear and cannot be described by a
Triangular Stress Diagram.
2 The Section Designed is Uneconomical as the Strength of the material
beyond the Elastic Zone is not taken into consideration.
3 Since Factor of Safety is on the Stresses under Working Loads; its not
possible to account for uncertainties associated with different types of loads
LIMIT STATE DESIGN METHOD
The object of Design based on Limit State Concept is to:
“Achieve an acceptable Probability that a Structure will not become
Unserviceable in its Life Time for the use for which it is intended, i.e. it will
not Reach to a Limit State.
A Structure with appropriate degrees of Reliability should be able to
withstand safely all the loads that are liable to act on it throughout its life and
it should also satisfy the Serviceability Requirements, such as,
Deflections and
Cracking
It should also be able to maintain the required Structural Integrity during
and after accidents, such as:
Fires
Explosions
Local Failures
In other words, all relevant Limit States must be considered in the Design to
ensure an adequate degree of “Safety and Serviceability”
Various Limit States, which needs to be examined are:
1Limit State of Collapse
The Limit State that concerns with the safety of people and/or safety of
structure are classified as “Limit State of Collapse” or Ultimate Limit State.
Following Limit States shall be verified where they are relevant
Loss of Equilibrium of the Structure or any part of it considered as a
Rigid Body
Failure due to Excessive Deformations, Rupture and Loss of Stability of
Structure.
Failure caused by Fatigue or other Time Dependent effects.
These above Limit States Corresponds to Maximum Load carrying capacity
Violation of Collapse Limit State implies Failure in the sense that a clearly
defined Limit State of Structural Usefulness has been exceeded.
However it does not mean a complete collapse
The Collapse Limit State may correspond to
Flexure
Compression
Shear
Torsion
2 Limit State of Serviceability
The Limit State that Concerns with the :
Functioning of the Structure or Structural Members under Normal use
Comfort of People orAppearance of the Construction work
Shall be Classified as Serviceability Limit State
The verification of Serviceability Limit State is based on the following aspects
(a) Deformations that affects
Appearance
Comfort of Users
The Function of the Users
(b) Vibrations
That Causes Discomfort
That Limits the Functional Effectiveness of Structure
(c) Damage/Cracking That is likely to affect
Appearance
Durability
Functioning
Hence effectively it is a S3problem i.e. for structure to be safe, it should have
1. Strength
2. Stiffness/Serviceability
3. Stability against
Overturning
Sliding
Shrinkage
And off course
4 Economy
The Limit State approach of Design basically considers the strength of material beyond
Elastic Limit where the Stress is no longer Proportional to Stress.
It follows a Parabolic variation.
This is due to the Non-Linear behaviour of concrete as it is a Heterogeneous Material.
Figure 3 Stress Strain Curve in Limit State Design
The Limit State of Design of RCC Structures takes into account the
probabilistic and Structural Variations in the
Material properties,
Loads and
Safety Factors.
Hence the Resistance to Bending ,Shear,Torsion and Axial Loads at every
section shall not be less than the Appropriate value at that section produced
by the probable most unfavourable combination of Loads on the Structure
using the appropriate Partial Safety Factors.
The Limit State of Collapse can be expressed by the following expression:
Where
R:- Resistance or capacity of structure
L:- Loads acting on the Structure
:- Load Factor >1
: Safety Factor associated with the uncertainties in the material
properties
The Factors and are called partial safety factors.
Characteristic & Design Values and Partial Safety Factors
In order to keep the Structure Safe, Partial Safety Factors are applied to the
calculations of Loads and Material Strength.
Characteristic Strength of Material
It means that value of Strength of Material below which not more than 5% of
test results are expected to fall.
The Characteristic Strength should be in accordance with Table 2 of IS 456
2000
Design Value
The Characteristic Strength is determined in a laboratory by testing 15 cm
cubes in compression under controlled conditions.
But at site the conditions are not controlled.
The strength is a non-quantified quantity and due to the Errors,Mistakes and
Ignorance of workers at site, it is possible that the required strength may not be
achieved.
Hence, in order that the structure does not fail due to lack of strength of
material the Design Strength is taken into consideration.
Design Strength =
Design Strength =
Where, fck :- Characteristic Strength of Material
m:- Partial Safety Factor appropriate to Material
The Partial Safety Factor for Material Strength is a DIVIDING Factor i.e. the
Material Strength is reduced and only corresponding loads are allowed on the
structure.
Partial Safety Factor m[as per IS 456 2000 clause 36.4.2.1] for
Concrete = 1.5
Steel = 1. 15
Characteristic Loads [Clause 36.2 of IS 456 2000]
The term Characteristic Load means that value of Load which has a 95% Probability
of not being exceeded during the life of the structure.
For the purpose, of determining the loads the loads are taken as per ::
Dead Loads as given in IS 875 [Part 1]
Imposed Loads as given in IS 875 [Part 2]
Wind Loads as given in IS 875 [Part 3]
Snow Loads as given in IS 875 [Part 4]
Seismic Loads as given in IS 1893
Shall be assumed as Characteristic Loads
Design Values
In order that the Loads coming on the Structure never surpasses or are more
than the Characteristic Loads, the value of the load given in the code is
increased by multiplying it by a factor known as Partial Safety Factor and is
termed as “Designed Load”
Design Load Fdis given by:
Where F:- Characteristic Load
f:- Partial Safety Factor appropriate to nature of loading
The Values of Partial Safety Factor for Load shall be taken as given in Table 18
of IS 456 2000 i.e.
Load
Combination
Dead Load
(DL)
Imposed Load
(IL)
Wind Load
(WL)
DL + IL 1.5 1.5 -
DL + WL 1.5 - 1.5
*DL + WL 0.9 - 1.5
DL +IL + WL
1.2 1.2 1.2
* When Stability against overturning is critical or Stress Reversal is
Critical
Stress Strain Relationship for Concrete
Fig 4 Idealized Stress Strain Curve and Stress Block Parameters for
Concrete
The Mechanical Properties of Concrete and therefore, the Stress Strain
Relationship is influenced by a number of factors such as:
Type of Aggregate and Concrete Mix
Strength of Concrete
Age of Concrete
Shape and Size of Concrete Specimen
Creep
The experimental Stress-Strain curve for concrete is too complicated to be
used in design.
The Code IS 456 2000 has idealized it as shown in Fig 4 [Given at page 69,
Clause 38, Fig 21].
Since the concrete is heterogeneous, the curve is a Parabola i.e. Stress is not
Proportional to Strain.
As per the Code the Curve remain parabolic upto a strain of 0.002, beyond
which the strain remains constant with increasing load until a strain of 0.0035
(0.35%) has reached when the concrete is said to have failed.
Stress Strain Relationship for Steel
Fig 5 Idealized Stress Strain Curves for Steel
Actual Stress Strain
curve for Mild Steel Idealized Stress Strain curve for
Steel having definite Yield Point
Actual Stress Strain
curve for HYSD Bars Idealized Stress Strain curve for
HYSD Bars
Fig 6 Typical Stress Strain Curves for Mild Steel and HYSD Bars
SINGLY REINFORCED SECTIONS
A
ATension
Compression
Clear Span
b
Dd
A Reinforced Concrete Flexure Member should be able to resist to the following
stresses induced due to Imposed Loads :
Tensile Stress
Compressive Stress
Shear Stress
Concrete :-
Fairly Strong in Compression
Weak in Tension
Tensile Strength taken as Zero
Steel:-
Very Strong in Tension
Steel takes up Tension in the Tensile Zone of the Flexural Member.
While Designing a Reinforced Concrete Section, the Loading,Span,Grade of
Concrete, Grade of Steel and Width of Section are usually known in advance.
The Section Dimensions and Area of Steel Bars [Reinforcing Steel] are to be
determined.
There can be no unique section for a given set of forces. There are many
possible combinations.
Thus the cost will decide the final design
ASSUMPTIONS
Design for the Limit State of Collapse in Flexure shall be based on the assumptions
as per IS 456 2000; Clause 38.1, p 69
Plane Sections Normal to the Axis remain Plane after Bending
The Maximum Strain in Concrete at the outermost Compression Fibre is taken as
0.0035 in Bending
The Relationship between Stress-Strain distribution in Concrete is Parabolic upto
a Strain of 0.002 and then constant upto a Strain of 0.0035 at which the concrete is
said to have failed [IS 456 2000, Fig 21, pp. 69]
For Design purpose the Compressive Strength of Concrete is taken as 0.67 times
the Characteristic Strength of Concrete. A Partial Safety Factor m= 1.5 shall be
applied in addition to this.
The Tensile Strength of Concrete is ignored
The Stress in the Reinforcement is derived from the representative Stress-Strain
Curve for the type of Steel used. The typical Curves are given in Fig 23 of IS 456
2000;pp.70
For Design Purposes the Partial Safety Factor of m= 1.15 shall be applied to the
Characteristic Strength of Steel
The Maximum Strain in Tension Reinforcement in the Section at Failure should
not be less than the following i.e.
Bending of Beams
We know that
OR
fcr = 0.7fCK
MOMENT OF RESISTANCE
Neutral Axis
Fig 8: Stress Block Parameters
Consider a Simply Supported Beam subjected to Bending under factored loads.
For Equilibrium total force of Compression must be equal to the total force of
Tension i.e. C = T
The applied Bending Moment at Collapse i.e. Factored Bending Moment is equal to
the Resisting Moment of the Section provided by the Internal Stresses.
This called the Ultimate Moment of Resistance.
Now Force = C = C1+ C2
C = Force x Area
And MR= Force x Lever Arm
The portion above the Neutral Axis is in Compression and the Strain is
proportional to distance from Neutral Axis (NA) to the Extreme Compression
Fibre i.e. Zero at the NA to a Maximum at the Extreme fibre.
The cross section below the NA is in Tension and hence the Concrete is
assumed to have Cracked.
All the Tensile Stresses are supposed to be borne by steel bars and stresses in
all the steel bars are equal.
The resultant Tensile Force thus acts at the Centroid of the Reinforcing Bars.
The distance from the Extreme Compression Fibre to the centroid of the
Reinforcing Bars i.e. line of action of Tensile Force is called the Effective Depth
'd'.
Now,
Maximum Compressive Stress in Concrete without Safety Factor
= 0.67 fCK [Assumption 4]
Let,
X1: Depth of Parabolic Portion
X2: Depth of Rectangular Portion
From,
Similar Triangles of Strain Diagram,
Depth of Parabolic Portion is
Or,
Depth of Rectangular Portion
X2=XU- X1
=
OR
Force of Compression
Parabolic Block :-
C1= Stress x Area
= (0.67 fCK ) ×(2/3 X1. b)
= (0.67 fCK ) ×(2/3 . XU. b)
C1= 0.255 fCK .b . XU
Rectangular Block:-
C2= Stress x Area
= (0.67 fCK ) ×( X2. b)
= (0.67 fCK ) ×( XU. b)
C2= 0.287 fCK .b . XU
Hence Total Force of Compression without Partial Safety Factor
CO= C1 + C2
= 0.255 fCK .b . XU+ 0.287 fCK .b . XU
CO= 0.542 fCK b XU{Without Partial Safety Factor}
Now Applying Partial Safety Factor of 1.5 the Design Force of Compression is:-
C = 0.36 fCK b XU
Now,
Moment of Resistance = Force ×Lever Arm
Lever Arm = Z = d ˗ a
Where,
'a' is the distance of line of action of force of compression from the extreme top
fibre.
To determine 'a' take moment of all the forces about top extreme fibre. i.e.
CO ×a = 0.225 fCK .b . XU2
a = 0.42 XU
Where,
XU: Depth of Neutral Axis from Top Fibre
B : Width of the Section
DEPTH OF NEUTRALAXIS
Depth of NA can be obtained by considering the equilibrium of normal force i.e.
Force of Compression = Force of Tension
Resultant Force of Compression
C = Average Stress ×Area
C = 0.36 fCK b XU
Resultant Force of Tension
T = 0.87 fYAt
Now
C = T
0.36 fCK b XU = 0.87 fYAt
OR,
Where,
At= Area of Tension Steel
LEVER ARM
The forces of Compression and Tension forms a Couple.
The distance between the lines of action of these two forces is called the Lever
Arm and is denoted by 'Z'.
The equation of equilibrium ΣM = 0 is satisfied by equating the factored
Bending Moment to the Moment of Resistance offered by either Force of
Compression or Force of Tension.
Lever Arm Z = d ˗a
OR
Z = d ˗0.42 XU
Now,
Moment of Resistance
Now,
Moment of Resistance w. r. t. Concrete
MRC = Compressive Force ×Lever Arm
MRC = 0.36 fCK b XU. Z
Moment of Resistance w. r. t. Steel
MRt = Compressive Force ×Lever Arm
MRt = 0.87 fYAt. Z
MODES OF FAILURE
Balanced Section :-
If the ratio of Steel to Concrete in a beam is such that the maximum strain in
concrete and steel reach simultaneously, a sudden failure would occur with less
alarming deflections.
Such a beam is referred to as a Balanced Reinforced Beam.
Under Reinforced Beam :-
When the amount of steel is kept less than that in the Balanced Section, the NA
moves upwards so as to reduce the area under compression to maintain the
Equilibrium Condition i.e.
Force of Compression is equal to the Force of Tension. [This is because the Force
of tension becomes less than the Force of Compression and hence the Force of
Compression has to be reduced]
In this process the Centre of Gravity of compressive forces also shifts upwards.
Under increasing Bending Moments Steel is strained beyond Yield Point and the
Maximum Strain in concrete remains less than 0.35% i.e. 0.0035.
If the beam is further loaded, the strain the section increases. Once the steel has
yielded it does not take any additional stresses for the additional strain and the
total force of tension remains constant.
However compressive stresses in concrete increases with the additional strain.
Thus the NA and Centre of Gravity of Compressive Forces further shifts
upwards to maintain Equilibrium.
This results in an increase in the Moment of Resistance of the Beam. This
process of shift in the NA continues until maximum strain reaches its Ultimate
Value i.e. 0.35% and the Concrete Crushes.
Such a beam is referred to as "Under Reinforced Beam".
The Failure is called Tension Failure because Yielding of Steel was responsible
for higher strains in concrete resulting in its failure.
Over Reinforced Beam :-
When the amount of steel is kept more than that in the Balanced, the NA tends
to move downwards and the Strain in Steel remains in Elastic Region.
If the beam is further loaded the stresses in steel keeps on increasing and so the
force of tension.
Here the force of tension is more than that of compression, and hence to
maintain the equilibrium of tensile and compressive forces the area of concrete
resisting compression has to increase so as to increase the force of compression.
In this process the NA furher shifts downwards until maximum strain in
concrete reaches its ultimate value of 0.35% and concrete crushes. The Steel is
well within Elastic Limits.
Such a beam is referred to as an "Over Reinforced Beam" and the failure as
Compression Failure
MAXIMUM DEPTH OF NEUTRAL AXIS
A compression failure in a Over Reinforced Beam is a Brittle Failure.
The Maximum Depth of NA is therefore limited to ensure that the Steel will
reach its Yield Point before Concrete fails in Compression, so that a brittle
Failure is avoided.
Let the Limiting Value of the depth of NA be XU Lim.
When,
XU= XU Lim [Balanced Section ]
XU< XU Lim [Under Reinforced Section ]
XU> XU Lim [Over Reinforced Section ]
The Limiting Value of Depth of NA XU Lim.for different grades of steel can be
obtained from Strain Diagram as shown in Fig 8.
From Similar Triangles
or
or
Where
E = 2 ×105N/mm2
The Limiting values of Depth of NA for different grades of steel are given in Table 1
From Fig 8
0.0035
0.002
X1
X2
XU Lim
d
Grade of Steel fY
(N/mm2)
XU Lim
250 0.53 d
415 0.48 d
500 0.46 d
550 0.44 d
Table 1 Maximum Depth of Neutral Axis
LIMITING VALUES OF TENSION STEELAND MOMENT OF RESISTANCE
Since the maximum depth of NA is limited the maximum value of moment of resistance
is also limited i.e.
MU Lim w. r. t Concrete = 0.36 fCK b XU. Z
MU Lim = 0.36 fCK b XU Lim (d ˗0.42 XU Lim ) {Balanced Section}
MU Lim w. r. t Steel = 0.87 fYAt. Z
MU Lim = 0.87 fYAt(d ˗0.42 XU Lim ) { Balanced Section}
For a given Rectangular Beam Section, the Limiting Values of MU Lim depends on
the Grade of Concrete and the Grade of Steel.
The Values of Limiting Moment of Resistance with respect to different Grades of
Concrete and Steel are given in Table 2.
Grade of
Concrete Fe 250 Fe 415 Fe 500 Fe 550
General 0.148fCK b d20.138fCK b d20.133fCK b d20.130fCK b d2
M 20 2.96 b d22.76 b d22.66 b d22.60 b d2
M 25 3.70 b d23.45 b d23.33 b d23.25 b d2
M 30 4.4 b d24.14 b d23.99 b d23.90 b d2
Table 2: Limiting Moment of Resistance (N-mm)
The percentage of Tensile Reinforcement Corresponding to Limiting Moment of
Resistance is obtained by equating the forces of tension and compression
0.87 fYAt= 0.36 fCK b XU Lim
Let,
The limiting values of tensile reinforcement percentage corresponding to different
grades of steel and concrete in a Singly Reinforced Beam are given in Table 3.
Grade of
Concrete
Percentage of Tensile Steel
fCK (N/mm2)
Fe 250 N/mm
2Fe 415 N/mm2Fe 500 N/mm2
Fe 550 N/mm2
M 20 1.76 0.96 0.76 0.66
M 25 2.20 1.19 0.94 0.83
M 30 2.64 1.43 1.13 1.00
Table 3: Limiting Tensile Steel in Rectangular Section (%)
Minimum and Maximum Tension Reinforcement
Minimum Reinforcement :[ As per Clause 26.5.1.1; pp. 46; of IS 456 - 2000 ]
The minimum area of tension reinforcement should not be less than that given
by the following :
Where,
AS= Minimum Area of Tension Reinforcement
b = Breadth of Beam or breadth of Web of T-Beam
d = Effective depth of Beam
fY= Characteristic Strength of Steel Reinforcement in N/mm2
Maximum Reinforcement :
The maximum area of tension reinforcement should not exceed 4% of the
Gross Cross-Sectional area of beam to avoid difficulty in placing and
compacting concrete properly in the formwork i.e.
ASM > 0.04 b D
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